For an analytic functional on , we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in . We determine the directions in which every solution can be continued analytically, by using the characteristic set.
@article{bwmeta1.element.bwnjournal-article-apmv74z1p105bwm,
author = {Ishimura, Ryuichi and Okada, Jun-ichi and Okada, Yasunori},
title = {Continuation of holomorphic solutions to convolution equations in complex domains},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {105-115},
zbl = {0962.46032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p105bwm}
}
Ishimura, Ryuichi; Okada, Jun-ichi; Okada, Yasunori. Continuation of holomorphic solutions to convolution equations in complex domains. Annales Polonici Mathematici, Tome 75 (2000) pp. 105-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p105bwm/
[000] [A] T. Aoki, Existence and continuation of holomorphic solutions of differential equations of infinite order, Adv. in Math. 72 (1988), 261-283. | Zbl 0702.35253
[001] [I1] R. Ishimura, A remark on the characteristic set for convolution equations, Mem. Fac. Sci. Kyushu Univ. 46 (1992), 195-199. | Zbl 0773.32006
[002] [I2] R. Ishimura, The characteristic set for differential-difference equations in real domains, Kyushu J. Math. 53 (1999), 107-114. | Zbl 0933.35200
[003] [I-O1] R. Ishimura and Y. Okada, The existence and the continuation of holomorphic solutions for convolution equations in tube domains, Bull. Soc. Math. France 122 (1994), 413-433. | Zbl 0826.35144
[004] [I-O2] R. Ishimura and Y. Okada, The micro-support of the complex defined by a convolution operator in tube domains, in: Singularities and Differential Equations, Banach Center Publ. 33, Inst. Math., Polish Acad. Sci., 1996, 105-114. | Zbl 0921.32003
[005] [I-O3] R. Ishimura and Y. Okada, Examples of convolution operators with described characteristics, in preparation.
[006] [I-Oj] R. Ishimura and Y. Okada, Sur la condition (S) de Kawai et la propriété de croissance régulière d'une fonction sous-harmonique et d'une fonction entière, Kyushu J. Math. 48 (1994), 257-263.
[007] [Ka] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 467-517. | Zbl 0212.46101
[008] [Ki] C. O. Kiselman, Prolongement des solutions d'une équation aux dérivées partielles à coefficients constants, Bull. Soc. Math. France 97 (1969), 329-356. | Zbl 0189.40502
[009] [Ko] Yu. F. Korobeĭnik, Convolution equations in the complex domain, Math. USSR-Sb. 36 (1985), 171-193.
[010] [Kr] A. S. Krivosheev, A criterion for the solvability of nonhomogeneous convolution equations in convex domains of , Math. USSR-Izv. 36 (1991), 497-517. | Zbl 0723.45005
[011] [Ll-G] P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Grundlehren Math. Wiss. 282, Springer, Berlin, 1986. | Zbl 0583.32001
[012] [Lv] B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monographs, Amer. Math. Soc., Providence, 1964.
[013] [R] L. I. Ronkin, Functions of Completely Regular Growth, Kluwer, 1992. | Zbl 0754.32001
[014] [Sé] A. Sébbar, Prolongement des solutions holomorphes de certains opérateurs différentiels d'ordre infini à coefficients constants, in: Séminaire Lelong-Skoda, Lecture Notes in Math. 822, Springer, Berlin, 1980, 199-220.
[015] [V] A. Vidras, Interpolation and division problems in spaces of entire functions with growth conditions and their applications, Doct. Diss., Univ. of Maryland. | Zbl 0842.32001
[016] [Z] M. Zerner, Domaines d'holomorphie des fonctions vérifiant une équation aux dérivées partielles, C. R. Acad. Sci. Paris 272 (1971), 1646-1648. | Zbl 0213.37004